Jornada SPM/CIM/CMAT - “Dia das Equaç˜oes” Relatório

Jornada SPM/CIM/CMAT - “Dia das Equações”
Relatório
Decorreu durante o dia 11 de Setembro de 2009
a Jornada SPM/CIM/CMAT - “Dia das Equações”.
Este evento realizou-se no Rio Douro, a bordo de um
barco rabelo e teve a participação de 29 matemáticos
entre os quais oito alunos de doutoramento. A organização esteve a cargo de Lisa Santos, Fernando
Miranda e Assis Azevedo do Centro de Matemática
da Universidade do Minho.
O programa da Jornada consistiu de cinco palestras cientı́ficas proferidas por Miguel Ramos (Universidade de Lisboa), Filipe Oliveira (Universidade Nova de Lisboa), Juha Videman (Instituto Superior Técnico),
Eugénio Rocha (Universidade de Aveiro) e José Miguel Urbano (Universidade de Coimbra) e de uma “mesa
redonda” moderada por José Francisco Rodrigues (Universidade de Lisboa).
Houve ainda uma palestra feita por Francisco Esteves, autor do livro “Vinhos do Douro”.
Programa Cientı́fico
• 09:45-10:20 - Boundedness and segregation of solutions for a Bose-Einstein type system, Miguel Ramos;
Abstract: We prove the existence of positive solutions for a system of the form
−∇u = u + u3 − β uv 2
−∇v = v + v 3 − β u2 v
in a smooth bounded domain of R3 , with Dirichlet boundary conditions, as well as their segregation
as the parameter β tends to infinity.
• 11:00-11:35 - On the Cauchy problem for the Zakharov-Schulman systems, Filipe Oliveira;
Abstract: In this communication we will consider the Zakharov-Schulman (Z-S) system in dimension
n=3
iut + L1 u = |u|2 u + uv
L2 v = L3 |u|2
where Lj denote spatial differential operators of order 2, with L1 non-degenerated and L2 elliptic.
This system is a model for the interaction of small-amplitude high-frequency waves with low-frequency
acoustic-type waves.
In dimension n = 2, by choosing L3 = ∂xx , (Z-S) reduces to the well-known Davey-Stewartson system,
which describes the propagation of water waves in one main direction with slow transverse modulation.
s
n
We will derive the local well-posedness of (Z-S) in Sobolevspaces
the energy space.
H (R ) below
it
The main tools are Strichartz-type estimates for the group e L1 t∈R and Lp commutator estimates
for fractional derivatives Ds , 0 < s < 1.
(Joint work with Jorge D. Silva and Mahendra Panthee.)
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• 11:35-12:10 - Existence of trapped modes in waveguides, Juha Videman;
Abstract: The interaction of linear water waves with totally or partially submerged obstacles is
considered in a homogeneous, inviscid, incompressible liquid. A sufficient condition for the existence
of localized trapped modes is established by introducing a trace operator that restricts the solutions to
the free surface. The modes correspond to localised solutions of a spectral problem, decaying at large
distances from the obstacles and belonging to the discrete spectrum below a positive cut-off value of
the continuous spectrum.
The sufficient condition is a simple relation between the cut-off value and some geometrical constants,
namely the surface integrals taken over the cross-sections of the submerged parts of the obstacles and
the line integrals along the parts of the free surface pierced by the obstacles.
Several particular cases are considered and the results are extended to a two-layer fluid consisting of
two immiscible liquid layers of different densities and to the problem of existence of Rayleigh-Bloch
surface waves traveling along a periodic family of obstacles and edge waves guided by, and propagating
along, a periodic seashore.
A new and simple proof for the comparison principle, a method often used in proving existence of
trapped modes, is also presented.
• 13:45-14:20 - Elliptic equations with critical exponents and singular terms, Eugénio Rocha;
Abstract: The main issue of the presentation is to discuss the existence and multiplicity of solutions
of the (linear) Dirichlet problem
−∇u(x) − λ/|x|2 u(x) = |u(x)|2∗ − 2u(x) + µ|x|α−2 u(x) + f (x) in Ω with u ∈ H01 (Ω),
where 0 ∈ Ω ⊆ RN (N ≥ 3) is a bounded domain with smooth boundary, 2∗ := 2N/(N − 2) is the
Sobolev critical exponent, 0 ≤ λ < ((N − 2)/2)2 and f ∈ L∞ (Ω). Briefly, we will also point out the
main ideas for the study of the existence of positive solutions of the (nonlinear) Dirichlet problem
−∇p u(x) = β(x)u(x)−η + f (x, u(x))
in Ω with u ∈ W01,p (Ω),
which involves the combined effect of a singular term (η ≥ 0) and a (p − 1)-linear term f (x, u) near
+∞.
(These are joint works with J. Chen and with J. Chen and N. Papageorgiou, respectively.)
• 14:20-14:55 - p(x)-harmonic functions with unbounded exponent in a subdomain, José Miguel Urbano;
Abstract: We study the Dirichlet problem for the p(x)-Laplacian, in the case of a variable exponent
p(x) that is infinite in a subdomain D. The main issue is to give a proper sense to what a solution
is. To this end, we consider the limit of the solutions un to the corresponding problem when pn (x) =
min(p(x), n), in particular, with pn = n in D. Under suitable assumptions on the data, we find that
such a limit exists and that it can be characterized as the unique solution of a variational minimization
problem, which is, in addition, infinity-harmonic within D. Moreover, we examine this limit in the
viscosity sense and find the boundary value problem it solves.
(Joint work with Juan J. Manfredi and Julio D. Rossi.)
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• 15:00 - Round table: Discussion of open questions.
A mesa redonda foi coordenada por José Francisco Rodrigues e foi composta pelos oradores convidados.
Alguns dos tópicos abordados foram:
– palavras de encorajamento aos alunos de doutoramento presentes no encontro, com a referência de não confiarem plenamente em resultados publicados;
– a grande fonte de problemas que nos surjem
provêm das ciências mais aplicadas, sendo
portanto relevante que se criem em Portugal
equipas multidisciplinares;
– Miguel Ramos apresentou uma questão em
aberto concreta, referindo que outros problemas poderiam surgir do problema proposto;
– Márcia Scialom pediu ajuda aos participantes na resolução de uma questão concreta.
Agradecimentos
A organização gostaria de agradecer à Sociedade Portuguesa de Matemática e ao Centro Internacional de
Matemática o convite para organizar esta jornada bem como o apoio prestado na organização e divulgação
do evento. Agradecemos o apoio financeiro dado pelo Centro Internacional de Matemática, pelo Centro
de Matemática da Universidade do Minho e pelo Projecto FCT, UT-Austin/MAT/0035/2008, “Analysis of
Nonlinear Partial Differential Equations”.
Gostarı́amos também de agradecer aos oradores e a todos os participantes esperando que a Jornada
SPM/CIM/CMAT tenha correspondido à suas expectativas.
Braga, 1 de Outubro de 2009
Lisa Santos
Fernando Miranda
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Assis Azevedo